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NumGy_LV12.qmd
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---
title: "Numerical Simulation Methods in Geophysics, Part 12: Advection problems and the Finite Volume Method"
subtitle: "1. MGPY+MGIN"
author: "[email protected]"
title-slide-attributes:
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format:
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<script>
window.MathJax = {
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packages: {'[+]': ['physics']}
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---
# Recap
1. The Finite Difference (FD) method
* Poisson equation in 1D, look into 2D/3D
* diffusion equation in 1D, time-stepping
1. Solving the hyperbolic (acoustic) wave equation in 1D
1. The Finite Element (FE) method
* Poisson and diffusion equation in 1D
* (complex) Helmholtz equation in 2D for EM problems
* solving EM problems and computational aspects
## Solution in pyGIMLi
:::: {.columns}
::: {.column}
step by step
```
import pygimli.solver as ps
A = ps.createStiffnessMatrix(mesh, a)
M = ps.createMassMatrix(mesh, b)
f = ps.createLoadVector(mesh, f)
ps.assembleNeumannBC(b, boundaries)
ps.assembleDirichletBC(A, b)
ps.assembleRobinBC(A, boundaries, b)
u = ps.linSolve(A, b)
```
:::
::: {.column}
or shortly
```
bc = {'Dirichlet': {4: 1.0, 3: 0.0},
'Neumann': {2: 1.0}}
u = ps.solveFiniteElements(mesh, a, b, f, bc=bc,
t=times, c)
u = pygimli.physics.seismics.solvePressureWave(
mesh, velocities, times, sourcePos, uSource)
```
:::
::::
$$ c\pdv{u}{t}= \div(a\grad u) + b u + f(\vb r, t)$$
## Advection
Volume elements move with velocity $\vb v$ and take (advectio=move with)
$$\pdv{T}{t} \Rightarrow \pdv{T}{t} + \dv{z}{t}\pdv{T}{z}$$
in 3D $\vb{v}\cdot\grad{T}$ $\Rightarrow$ advection-dispersion equation
$$\pdv{T}{t} = \div a\grad{T} + Q - \vb{v}\cdot\grad{T}$$
## Advection-dispersion equation (general)
$$ \pdv{u}{t}+\vb v\cdot\grad u = \div(a\grad u) + b u + f(\vb r, t)$$
$\vb v\cdot\grad u$ (Advection), $\div(a\grad u)$ (Dispersion=Diffusion)
Solve, e.g., by [`pyGIMLi.solver.solveFiniteVolume` (Link)](https://www.pygimli.org/pygimliapi/_generated/pygimli.solver.html#pygimli.solver.solveFiniteVolume)
```
bc = dict(Dirichlet={4: 1.0, 3: 0.0}, Neumann={2: 1.0})
u = ps.solveFiniteVolume(mesh, a, b, f, bc=bc, t=times, c)
```
## Computational fluid dynamics
Magnetohydrodynamic induction equation
$$\pdv{\vb B}{t} = \curl \frac{1}{\mu\sigma} \curl \vb B + \curl(\vb v \cross \vb B)$$
Forces on fluid (induction, Coriolis, pressure, gravity, Lorentz)
$$ \rho \pdv{\vb v}{t} + \rho (\vb v \cdot \grad) \vb v + \grad p = -2\omega\cross\vb v+\alpha T \vb g + \vb j \cross \vb B / \rho $$
## FE solution for inhomogeneous viscosity
{fig-align="center"}
## Simple 1D advection problem
$$\pdv{u}{t} + \pdv{v(u)}{x}=0$$
Solution $u_i$ at node $i$ represents average value over cell
$$ \overline{u}_i(t) = \frac{1}{x_{i+1/2}-x_{i-1/2}} \int_{x_{i-1/2}}^{x_{i+1/2}} u(x, t)\dd x $$
$$ \int_{t_1}^{t_2} \pdv{u}{t} = u(x, t_2)-u(x, t_1) = - \int_{t_1}^{t_2} \pdv{v(u)}{x} $$
## FV
## Simple 1D advection problem
$$ u(x, t_2) = u(x, t_1) - \int_{t_1}^{t_2} \pdv{v(u)}{x} $$
$$ \overline{u}(t_2) = \frac{1}{x_{i+1/2}-x_{i-1/2}} \int_{x_{i-1/2}}^{x_{i+1/2}}\qty\Big(u(x,t_1) - \int_{t_1}^{t_2} \pdv{v(u)}{x}) $$
$$ \overline{u}(t_2) = \overline{u}(t_1) - \frac{1}{x_{i+1/2}-x_{i-1/2}}
\qty\Big(\int_{t_1}^{t_2} v_{i+1/2} \dd t - \int_{t_1}^{t_2} v_{i-1/2} \dd t) $$
## Result
$$ \pdv{\overline{u}_i}{t} +\frac{1}{\Delta x_i} (v_{i+1/2}-v_{i-1/2}) = 0 $$
::: {.callout-note}
Finite volume schemes are conservative as cell averages change through the edge fluxes. In other words, one cell's loss is always another cell's gain!
:::
## Visualization
{fig-align="center"}
## Similarity to staggered (E, B) grid methods
{fig-align="center"}
## Conservation law (3D) problem
$$\pdv{\vb u}{t} + \div\vb v(u)= 0$$
volume integral over cell, using Gauss' law
$$\int_{\Omega_i} \pdv{\vb u}{t}\dd\Omega+\int_{\Omega_i} \div\vb v(u) \dd\Omega = 0 =
\int_{\Omega_i} \pdv{\vb u}{t}\dd\Omega+\int_{\Gamma_i} \vb v(u) \cdot \vb n \dd{\Gamma} $$
$$\Rightarrow \pdv{\overline{u}}{t} +\frac{1}{V_i}\int_{\Gamma_i}\vb v(u)\cdot \vb n \dd{\Gamma} =0 $$